Recall , `pi ` is defined as the ratio of the circumference (say c) of a circle to its diameter (say d ) . That is `, pi= ( c)/( d) . ` This seems to contradict the fact that `pi ` is irrational . How will you resolve this contradiction ?

Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid-points of the chords of the circle C that subtend an angle of (2pi)/(3) at its centre is ....

Two identical balls A and B each of mass 0.1 kg are attached to two identical mass less is springs. The spring-mass system is constrained to move inside a right smooth pipe bent in the form of a circle as shown in figure . The pipe is fixed in a horizontal plane. The center of the balls can move in a circle of radius 0.06 m . Each spring has a natural length of 0.06 pi m and force constant 0.1 N//m . Initially, both the balls are displaced by an angle theta = pi//6 radians with respect to diameter PQ of the circle and released from rest. a. Calculate the frequency of oscillation of the ball B. b. What is the total energy of the system ? c. Find the speed of the ball A when A and B are at the two ends of the diameter PQ.

Answer carefully: (a) Two large conducting spheres carrying charges Q_(1) and Q_(2) are brought close to each other. Is the magnitude of electrostatic force between them exactly given by Q_(1),Q_(2)//4pi epsilon_(0)r^(2) , where r is the distance between their centres? (b) If Coulomb’s law involved 1//r^(3) dependence (instead of would Gauss’s law be still true ? (c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point? (d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical? (e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there? (f) What meaning would you give to the capacitance of a single conductor? (g) Guess a possible reason why water has a much greater dielectric constant (= 80) than say, mica (= 6).

In an experiment to estimate the size of a molecule of oleic acid 1 mL of oleic acid is dissolved in 19 ml of alcohol. Then I mL of this solution is diluted to 20 mL by adding alcohol. Now 1 drop of this diluted solution is placed on water in a shallow through. The solution spreads over the surface of water forming one molecule thick layer. Now lycopodium powder is sprinkled evenly over the film and its diameter is measured. Knowing the volume of the drop and area of the film we can calculate the thickness of the film which will give us the size of oleic acid molecule. Read the passage carefully and answer the following questions. (a) Why do we dissolve oleic acid in alcohol ? (b) What is the role of lycopodium powder ? (c) What would be the volume of oleic acid in each ml of solution prepared ? (d) How will you calculate the volume of n drops of this solution of oleic acid ? (e) What will be the volume of oleic acid in one drop of this solution ?

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x^(2)cos((pi)/(4)))/(sinx)

A transverse harmonic wave on a string is described by y(x,t) = 3.0 sin (36 t + 0.018x + pi//4) where x and y are in cm and t in s. The positive direction of x is from left to right (a) Is this a travelling wave or a stationary wave? If it is travelling what are the speed and direction of its propagation? (b) What are its amplitude and frequency? (c ) What is the initial phase at the origin ? (d) What is the least distance between two successive crests in the wave?

A circle, having radii 'a' has line charge distribution over its circumference having linear charge density lambda = lambda_(0)cos^(2)theta . Calculate the total electric charge residing on the circumference of the circle. [Hint: int_(0)^(2pi)cos^(2)d theta =pi ]

A bag contains a red ball, a black ball, a yellow ball and a white ball. One ball is selected randomly and note its colour and then put it into the bag. Again second ball is selected and note its colour. Write the sample space for this experiment. Hence, describe the following events : (1) A: Selected both the balls are of same colours. (2) B: Only one ball is white. (3) C: At least one ball is white. (4) D: Both the balls are of different colour. Hence, find AnnB,BuuC,AuuD,AnnD . What can we say about the events B and C ? What can we say about the events A and D?